NAG Library Function Document

nag_dgtcon (f07cgc)

void	nag_dgtcon (Nag_NormType norm, Integer n, const double dl[], const double d[], const double du[], const double du2[], const Integer ipiv[], double anorm, double rcond, NagError fail)

3 Description

nag_dgtcon (f07cgc) should be preceded by a call to nag_dgttrf (f07cdc), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix

A

A = P L U,

where

P

is a permutation matrix,

L

is unit lower triangular with at most one nonzero subdiagonal element in each column, and

U

is an upper triangular band matrix, with two superdiagonals. nag_dgtcon (f07cgc) then utilizes the factorization to estimate either

{‖A^{- 1}‖}_{1}

{‖A^{- 1}‖}_{\infty}

, from which the estimate of the reciprocal of the condition number of

A

1 / κ (A)

is computed as either

1 / κ_{1} (A) = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})

1 / κ_{\infty} (A) = 1 / ({‖A‖}_{\infty} {‖A^{- 1}‖}_{\infty}) .

1 / κ (A)

is returned, rather than

κ (A)

, since when

A

is singular

κ (A)

is infinite.

Note that

κ_{\infty} (A) = κ_{1} (A^{T})

4 References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5 Arguments

1: norm – Nag_NormTypeInput

On entry: specifies the norm to be used to estimate

κ (A)

$norm = Nag_OneNorm$: Estimate $κ_{1} (A)$ .
$norm = Nag_InfNorm$: Estimate $κ_{\infty} (A)$ .

Constraint:

norm = Nag_OneNorm

Nag_InfNorm

2: n – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: dl[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array dl must be at least

\max (1, n - 1)

On entry: must contain the

(n - 1)

multipliers that define the matrix

L

of the

L U

factorization of

A

4: d[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array d must be at least

\max (1, n)

On entry: must contain the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

5: du[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array du must be at least

\max (1, n - 1)

On entry: must contain the

(n - 1)

elements of the first superdiagonal of

U

6: du2[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array du2 must be at least

\max (1, n - 2)

On entry: must contain the

(n - 2)

elements of the second superdiagonal of

U

7: ipiv[ $\dim$ ] – const IntegerInput

Note: the dimension, dim, of the array ipiv must be at least

\max (1, n)

On entry: must contain the

n

pivot indices that define the permutation matrix

P

. At the

i

th step, row

i

of the matrix was interchanged with row

ipiv [i - 1]

, and

ipiv [i - 1]

must always be either

i

(i + 1)

ipiv [i - 1] = i

indicating that a row interchange was not performed.

8: anorm – doubleInput

On entry: if

norm = Nag_OneNorm

, the

1

-norm of the original matrix

A

norm = Nag_InfNorm

, the

\infty

-norm of the original matrix

A

anorm may be computed as demonstrated in Section 10 for the

1

-norm. The

\infty

-norm may be similarly computed by swapping the dl and du arrays in the code for the

1

-norm.

anorm must be computed either before calling nag_dgttrf (f07cdc) or else from a copy of the original matrix

A

(see Section 10).

Constraint:

anorm \geq 0.0

9: rcond – double *Output

On exit: contains an estimate of the reciprocal condition number.

10: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL: On entry, $anorm = ⟨value⟩$ .
Constraint: $anorm \geq 0.0$ .

7 Accuracy

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

8 Parallelism and Performance

nag_dgtcon (f07cgc) is not threaded by NAG in any implementation.

nag_dgtcon (f07cgc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The total number of floating-point operations required to perform a solve is proportional to

n

The complex analogue of this function is nag_zgtcon (f07cuc).

10 Example

This example estimates the condition number in the

1

-norm of the tridiagonal matrix

A

given by

A = (\begin{array}{r} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{array}) .

NAG Library Function Documentnag_dgtcon (f07cgc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_dgtcon (f07cgc)